where the two normalization factors related to stellar winds and supernovae are respectively \(N_\ast =Z_{\ast ,i} - Z_i\)  (with \(Z_{\ast ,i}\)  the stellar abundance) and \(N_{\rm SN}=y_{{\rm SN},i}/(Z_{i,\odot } M_{\rm SN})\)  (with \(y_{{\rm SN},i}\)  the SN yield in \(M_{\odot }\)  and \(M_{\rm SN}\)  the ejected supernova mass).
The BGG stellar density \(\rho _\ast \)  is usually modeled via a de Vaucoulers profile [1]}.
The specific injection rates due to SWs and SN are
\(\alpha _\ast \approx 4.7\times 10^{-20}(t/t_{\rm now})^{-1.3}\)  s\(^{-1}\)  and
\(\alpha _{\rm SN} \simeq 3.2\times 10^{-20}\,r_{\rm SN}(t) (M_{\rm SN}/M_{\odot })\,\Upsilon _B^{-1}\ {\rm s^{-1}}\) , where \(\Upsilon _B^{-1}\)  is the optical stellar mass-to-light ratio in the B band [2]}, [3]}, [4]}, [5]}, respectively.
As introduced above, iron is one of the best metals to leverage as dynamical tracer in HR simulations: in BGGs/ETGs (which have star-formation histories peaked at early times) SNII are mostly consumed at high redshifts, while SNIa – exploding in binary systems with a white dwarf – drive the BGG long-term iron enrichment. The local SNIa rate is \(r_{\rm SNIa} \sim 0.1 (t/t_{\rm now})^{-1.1}\)  SNU (per 100 year and stellar luminosity \(10^{10}\,L_{\rm B,\odot }\) ; e.g. [3]}, [4]}, [5]}). The iron normalization values for the SNIa (\(M_{\rm SNIa} = 1.4\,M_{\odot }\) ) are \(y_{\rm SNIa, Fe} = 0.7\,M_{\odot }\sim 10\,y_{\rm SNII, Fe}\)  and \(Z_{{\rm Fe},\odot } = 1.83\times 10^{-3}\) .
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